On the Number of Figures in the Period of the Reciprocal of Every Prime Number below 20,000

200 Mr. W. Shanks on the Period of the [Feb. 19,

III. ^^ On the Number of Figures in the Period of the Reciprocal
of every Prime Nu^mber below 20^000/^ By William
Shanks. Communicated by the Rev. George Salmon.
Received December 2^ 1873^.

The following Table, in reality the joint production of the Rev. George
Salmon, F.R.S., and myself, was commenced, and indeed nearly com-
pleted, before either calculator was distinctly aware that Burckhardt,
Jacobi, or Desmarest had written or published any thing on the same
subject. This fact is perhaps to be regretted ; but it has led to the in-
dependent recalculation, by two different methods, both of Burckhardt's
( Jacobi's Table is professedly a reprint of Burckhardt's) and of Desmarest's
Table, and has resulted in the detection of several errors, which have, as
far as I know, never before been pointed out. These errors, in the first
place regarded as discrepancies, have been carefully examined ; in fact
every case has been reworked by me, with the view of either proving or
disproving the accuracy of such numbers as differ from those in our
Table. The result is, that such discrepancies are found to be errors both
in Burckhardt and Desmarest. The two lists of errors are given below.

I now proceed to give the theorems used, and some account of the
means employed by me in forming the Table.

Let P be any prime number, except 2 and 5. Then, from Fermat's

:theorem, we have — pr~^l ; or, adopting the usual notation, 10^ ^^1.

Again, since the number of figures in the period of the reciprocal of

all primes is not P~l (or, in other words, since 10 is not a primitive

root of all primes),
p-i
Let 10 » ^1, where n is even or odd, not less than 2, and not greater

P~l

than — o — Then we have

(1) The number of figures in the period of the reciprocal of P is either
P— 1 or a submultiple of P -- 1, -i qo

(2) Let a and h be integers, and let m be the remainder from -p ;
that is, let 10«=m; then lO'^^—m^

In practice h is never greater than 2, at least little or no advantage is

P-1

gained by putting h higher. Also ah need not be greater than — ^—•

P-1 .

Cor, "When m is greater than —^ — we may obviously use P — m, or

simply — m; for (P — m)^=P^— 2Pm+w^^wS or, because (— m)^==m^,
h being 2.

(3) Let lO'^—m, and 10*^n; then 10«+*=mw.
In practice a-^-h is never greater than P — 1.

Got. 1. When m and n are each of them less than P, we may with
advantage use ~m and — n; that is, we may subtract m and n severally
from P ; for (P — m)(P — n) = P^ — V{m -\- n) -f- mn == ( — m)( -~ n),

P-] , P-1

Cor, 2. "When m is > — ^ ^ and n is < — o~~ ? or vice versa, we may use

* The part from 17,000 to 20,000 was reoeived January 8, 1874.

1874.] Reciprocal of every Prime Number below 20^000. 201

— m and n, or vice versa, obtaining a negative result, which becomes
positive by being subtracted from P. p__^

(4) Let 2c and Sc, not greater than - " ^ ? be submultiples of P— -1 ;

and let 10^= + S, and lO^'^^S ™ 1 ; then 10'^''— + 1. This is evident from
(2) and (3). ^

Prom (1) we have 10 ^ ^= + 1, according as the submultiple of P — 1 is
even or odd.

On these theorems and adjuncts my calculations have been based.

They enable us to find the remainder either from ^ , or from any

P-l -L

submultiple, such as — p— , or from any figure in —p—- , and, if required,

the jflgiire itself. Compared with other methods, such for instance as
Dr. Salmon's *, mine may seem tedious, requiring as it does much multi-
plication and division. All I can say is, I did not find it so, though I
am free to admit that the calculation of such a Table as ours demands
very considerable labour.

It would be foreign to my purpose to enter upon the consideration of
primitive roots, or even of prime numbers. If we have found 10 to be a
primitive root of a great many prime numbers between 10,000 and
20,000, we have contributed something, as far as I know, quite new.
In addition to this we have found the number of figures in the period of
each of the other primes between 10,000 and 20,000, and have corrected
upwards of 70 errors in Burckhardt's and Desmarest's Tables.

I beg to refer to the w^orks of Euler, Lagrange, Legendre, Gauss,' Poinsot,
Cauchy, and Jacobi (mentioned by Desmarest), and to Desmarest himself,
for valuable information touching prime numbers and primitive roots.

I cannot, however, refrain from quoting from Desmarest's ' Theorie des
ISTombres ' the view of Euler as to prime numbers and primitive roots ; —
" On ne pent saisir entre un nombre premier et les racines primitives qui
lui appartiennent, aucune relation d'oii Ton puisse deduire ime seide de
ces racines, de sorte que la loi qui regne entre elles parait aussi profonde-
ment cachee que celle qui existe entre les nombres premiers eux-memes.

ISTot discouraged by Euler's remark, Desmarest thus writes : — " Car
pourquoi nous serait-il defendu d'ajouter que nous croyons que I'intel-
ligence humaine n'a pas, sur ce point, dit son dernier mot, et que les
operations nombreuses que nous avons dii faire sur les nombres, ne nous
ont pas convaincu de I'impossibilite de saisir, sinon I'ensemble, du moins
quelques-uns des anneaux de la chaine mysterieuse qui unit les racines
primitives aux nombres premiers/'

'''' Note hy Dr, Salmon. — The method here referred to is explained, ' Messenger of
Mathematics' (1872), p. 49. It is founded on the remark that if we have 10'^== 2?,
]0*^s2'?, we may deduce lO"*?-^^^!. Thus, let the prime be 251, we can at once write
down the equations 103£EE-2^ 2«^10\ whence immediately IO'^-^eb-I, lO^^:^!.
In like manner from the equations 10«iEEE2^3?', 10*=2'»3'?, 10;^2'*3'-, we deduce that
the number of figures in the period of the reciprocal of the prime is

a{onr-~')ig)-\'h{np—Tl)-^c{lq—m2)).

By the application of these principles I calculated the results obtained in the fol-
lowing Table as far as 18500. For the primes above that number Mr. Shanks is solely
responsible ; but my experience of his accuracy giyes me confidence in his results,

VOL. XXTI, Q

.V

Mr, W. Shanks on the Period of the [Feb. 19^

Table I. List of Errors in Desmarest's Table.

Primes.

3 \
omitted j

277

317

397

449
787

1409

1657

1733
1889

2087

3253

3373

3413

3517

3541

3547

3637
3677

3769
3821

3911
4049

4167

should be

4157

4397
4621

4651

5871

should be

4871

4943
5081

5107
5407
5479
5519

No. of figures

in period of

reciprocal.

omitted

138
158
198
224
786
1408
276

433
59

P-

8

1 „ P-1

9

271
562

853

1758

60

3546
3636

919

942

1910

3910

1012

should be

157
4620

2325

2471

2540

5106

901

5478
5518

55

J5
}>
}J
>5
}>

J}

>}

JJ
}}
35
JJ
3>
5J
}J
J5
55
JJ
JJ
??

1^0. of

figures in

jjeriod of

reciprocal.

5?

55
JJ
55
JJ

J)

55

69

79

99

32

393

32
552
866

118
298

542

843
1706

879

20

1773
909

1838

1884

3820

1955

2024

314

924

4650

4942
1270

2551
1802

2739
2759

Primes.

No, of figures
in period of
reciprocal.

8423

8521
8609
8681
8893
8999
9067
9187

9397
952X

9629

9649
9941

1389

577.8
582

3050
313S
3143

6780

3498

3500
509

3740
3780

3858
25S0

392

omitted

T335

4043

809a
8100
4109

/ 8422 1
1 421 1 j

355
538
4340
4446
8998
9066
9186
4698
952

4814
1206
9940

should be

JJ

JJ

JJ

JJ

JJ

55
JJ
JJ
JJ
JJ
JJ
JJ
JJ
JJ

JJ

>j
5?
J?
}}
j>

JJ

ij

fj
JJ

S5
JJ
JJ
J1
JJ
JJ
JJ
JJ
JJ

No. of

figures in

X)eriod of

reciprosal.

926

2889

2913

1220

1569
6286

1356
1749
1750
1018

748
1890
1929

860
56

3926

2670
8086
4046
1620
8218

8422

710

1076

868

2223

4499

4533

4593
81

595
9628

603
1988

]S[,]3. There are 64 errors, 3 misprints, 2 omissions, viz. 3 and 7853

Table II. Lis } 0:: Errors in Burc.iharc.t's anc. Jacobi's

Tables.

No. of figures

No. of figures

Primes.

in period of

in period of

reciprocal.

recixDroeal.

Burckhardt and Jacobi ...

911

450

should be

455

Burcldiardt

1979

1976

JJ

1978

Jacobi

3462
should be •

3467

i

/

1213

I2I2

202 i

1597

266

*,

133

1831

9^5

305

Burckhardt and Jacobi. . . -<

1951

390

395

1993 1992

664

231 1 I ^.6z

231

2437 ; 2436

I218

, ^ ^ _; \

367 • 3466

^ *

1733 1

N,B. Ther«^ are 3 misprints and 8 errors.

1874.] Reciprocal of every Prime Number beloiv SO^OOO. 303

d

ht-hand columns, imme-
reciprocal of each prime.

Table III.

3

7
II

13

17
19
23
29

31

37
41

43

47

53

59
61

67

71

73
79
83
89

97
101

103

107

109

113

127

131

137

139
149

151

157
163

167

173

179
181

191

193
197

199
211

223

227

229

233

^39

241

251

257
263

269

271

277

281

283

293

307

I

6
2
6
16
18
22
28

15

3

5

21
46

13

58
60

33

35
8

13
41

44

96

4
34

53
108

112
42

130

8

46

148

75
78

81

166

43
178
i3o

95
192

98

99

30
222
113

228

232

7
30

50

256
262
268

5
69

28

141

146

153

311

155

691

230

1 109

1108

1567

313

312

701

700

1117

558

1571

317

79

709

708

1123

561

1579

331

110

719

359

1129

564

1583

337

336

727

726

1151

575

X597

347

173

733

61

1153

1152

1 601

349

116

739

246

1163

581

1607

353

32

743

742

1171

1 1 70

1609

359

179

751

125

Ii8i

1180

1613

367

366

757

27 •

1187

593

1619

373

186

761

380

"93

1192

1621

379

378

769

192

1201

200

1627

^l^

382

773

193

1213

202

1637

389

388

787

393

1217

1216

1657

397

99

797

199

1223

1222

1663

401

200

809

202

1229

1228

1667

409

204

811

810

1231

41

1669

419

418

821

820

1237

206

i%3

421

140

823

822

1249

208

1697

431

215

827

413

1259

1258

1699

433

432

829

276

1277

638

1709

439

219

839

419

1279

639

1721

443

221

853

213

1283

64 1

1723

449

32

857

856

1289

92

1733

457

152

859

26

1291

1290

1 741

461

460

863

862

1297

1296

1747

463

154

877

438

1301

1300

1753

467

233

881

440

1303

1302

1759

479

239

883

441

1307

653

1777

487

486

887

886

1319

659

1783

491

490

907

151

1321

55

1787

499

498

911

455

1327

1326

1789

503

502

919

459

1361

680

i8oi

509

508

929

464

1367

1366

1811

521

5^

937

936

1373

686

1823

523

261

941

940

1381

1380

1831

541

540

947

473

1399

699

1847

547

91

953

952

1409

32

1861

557

278

967

322

1423

158

1867

563

281

971

970

1427

713

1871

569

284

977

976

1429

1428

1873

571

570

983

982

1433

1432

1877

577

576

991

495

1439

719

1879

587

293

997

166

1447

1446

1889

593

59^

1009

252

145 1

290

1901

599

299

1013

253

H53

726

1907

601

300

1019

ioi8

H59

162

19x3

607

202

1021

1020

147 1

735

1931

613

51

1031

103

148 1

740

1933

617

88

1033

1032

1483

247

1949

619

618

1039

5x9

1487

i486

1951

631

315

1049

524

1489

248

1973

641

32

105 1

1050

1493

373

1979

643

107

106 1

212

1499

214

1987

647

646

1063

1062

1511

755

1993

653

326

1069

1068

1523

761

1997

659

658

1087

1086

1531

1530

1999

661

220

1091

1090

1543

154^

2003

673

224

1093

273

1549

1548

2011

677

338

^097

1096

1553

155a

2017

683

341

U03

1102

1559

779

2027

1566

1570
1578

1582

133

200
1606

201

403
I6I8

1620
271

409

552

1662

833
556

423

1696

566

1708

430

287

866

1740

291

584

879

1776

1782
893

1788
900

1810

1822

3<>5
1846

1 860

933

935

1872

938

313
118

380

953
1912

386

21

1948

195
986

1978

331
664

998

999

looi

670
2016
1013

Q

9.

301.

Mr. W. Shanks on the Period of the

[Feb. 19,

Table III. {continued).

2029

2028

2539

2538

3023

3022

3547

1773

4057

4056

2039

IOI9

2543

2542

3037

253

3557

254

4073

4072

2053

342

2549

2548

3041

380

3559

1779

4079

2039

2063

2062

2551

425

3049

508

3571

3570

4091

4090

2069

2068

2557

639

3061

204

3581

3580

4093

22

2081

1040

2579

2578

3067

1533

3583

1194

4099

4098

2083

1041

2591

259

3079

1539

3593

3592

4H1

2055

2087

298

2593

2592

3083

1541

3607

3606

4127

4126

2089

1044

2609

1304

3089

1544

3613

602

4129

2064

2099

2098

2617..

2616

3109

148

3617

3616

4133

1033

2III

1055

2621

2620

3119

1559

3623

3622

4139

4138

2II3

2II2

2633

2632

3121

156

3631

1815

4153

4152

2129

532

2647

882

3^37

3136

3637

909

4157

2078

213I

710

2657

2656

3163

1581

3643

1821

4159

693

2137

2136

2659

886

3167

3166

3659

3658

4177

4176

214I

2140

2663

2662

3169

72

3671

367

4201

75

2143

2142

267Q

1335

3181

636

3673

3672

42H

4210

2153

2152

2677

223

3187

177

3677

1838

4217

4216

2161

30

2683

447

3191

29

3691

1230

4219

4218

2179

2178

2687

2686

3203

1601

3697

1232

4229

4228

2203

IIOI

2689

42

3209

1604

3701

3700

4231

2115

2207

2206

2693

1346

3217

1072

3709

3708

4241

1060

2213

553

2699

2698

3221

3220

3719

1859

4243

2121

2221

2220

2707

1353

3229

1076

3727

3726

4253

1063

2237

1118

271 1

1355

3251

3250

3733

933

4259

4258

2239

1119

2713

2712

3253

542

3739

1246

4261

4260

2243

1121

2719

1359

3257

3256

3761

1880

4271

2135

2251

2250

2729

682

3^59

3258

3767

3766

4273

1424

2267

1133

2731

2730

3271

1635

3769

1884

4283

2141

2269

2268

2741

2740

3299

3298

3779

3778

4289

2144

2273

2272

2749

916

3301

3300

3793

1264

4297

1432

2281

228

2753

2752

3307

1653

3797

949

4327

4326

2287

762

2767

2766

3313

3312

3803

190!

4337

4336

2293

1 146

2777

2776

3319

553

3821

3820

4339

4338

2297

2296

2789

2788

3323

1661

3823

1274

4349

4348

2309

2308

2791

31

3329

832

3833

3832

4357

242

2311

231

2797

699

3331

333^

3847

3846

4363

2181

2333

583

2801

1400

3343

3342

3851

770

4373

1093

2339

2338

2803

140 1

3347

1673

3853

963

4391

2195

2341

2340

2819

2818

3359

1679

3863

3862

4397

314

2347

1173

2833

2832

3361

1680

3877

969

4409

ss-^

2351

1175

2837

709

3371

3370

3881

1940

4421

4420

2357

1178

2843

142 1

3373

843

3889

1944

4423

4^22

2371

2370

2851

2850

3389

3388

3907

1953

4441

2220

2377

264

2857

408

3391

1695

3911

1955

4447

4446

2381

476

2861

2860

3407

3406

3917

195S

4451

4450

2383

2382

2879

1439

3413

1706

3919

653

4457

4456

2389

2388

2887

2886

3433

3432

3923

1961

4463

4462

2393

184

2897

2896

3449

431

3929

49!

4481

2240

2399

1 199

2903

2902

3457

384

3931

1310

4483

249

241 1

2410

2909

2908

3461

3460

3943

3942

4493

1123

2417

2416

2917^

1458

3463

3462

3947

1973

4507

751

2423

2422

2927

2926

3467

1733

3967

3966

4513

1504

2437

12l8

2939

2938

3469

3468

3989

3988

4517

2058

2441

305

2953

984

3491

698

4001

500

4519

753

2447

2446

2957

1478

3499

318

4003

87

4523

2261

2459

2458

2963

148 1

3511

1755

4007

4006

4547

2273

2467

137

2969

371

3517

879

4013

34

4549

1516

2473

2472

2971

2970

3527

3526

4019

4018

4561

2280

2477

619

2999

1499

3529

1764

4021

268

4567

4566

2503

278

3001

1500

3533

1766

4027

2013

4583

4582

2521

630

3011

3010

3539

3538

4049

2024

4591

2295

2531

46

3019

3018

3541

20

4051

4050

4597

2298

1874.] Recijjrocal of every Prime Number heloiv SO^COO. 205

Table III. {continued) »

4603

2301

5147

2573

5689

316

6247

6246

6803

3401

4621

924

5^53

5152

5693

1423

6257

6256

6823

6822

4637

6i

5167

5166

5701

5700

6263

6262

6827

3413

4639

2319

S^7^

no

5711

571

6269

6268

6829

6828

4643

2321

5179

5178

5717

1429

6271

1045

6833

6832

4649

7

5189

5188

5737

5736

6277

1569

6841

855

4651

4650

5197

433

5741

5740

6287

6286

6857

6856

4657

1552

5209

372

5743

5742

6299

94

6863

6862

4663

222

5227

2613

5749

5748

6301

6300

6869

6868

4673

4672

5231

2615

5779

28S9

6317

3158

6871

3435

4679

2339

5233

5232

5783

5782

6323

3161

6883

3441

4691

4690

5237

11

5791

965

6329

3164

6899

6898

4703

4702

5261

1052

5801

1450

6337

6336

6907

1151

4721

2360

5273

5272

5807

5806

6343

6342

6911

3455

4723

2361

5279

2639

5813

2906

6353

6352

6947

3473

4729

1182

5281

2640

5821

5820

6359

3179

6949

6948

4733

1183

5297

S296

5827

2913

6361

1590

6959

3479

4751

2375

5303

5302

5839

2919

6367

6366

6961

3480

4759

2^79

5309

5308

5843

2921

6373

1062

6967

6966

4783

4782

5323

2661

5849

1462

6379

2126

6971

6970

4787

2393

5333

1333

5851

1950

6389

6388

6977

6976

4789

228

5347

2673

5857

5856

6397

78

6983

6982

4793

4792

5351

2675

5861

5860

6421

2140

6991

3495

4799

2399

5381

5380

5867

2933

6427

1071

6997

1749

4801

800

5387

2693

5869

5868

6449

1612

7001

1750

4813

802

5393

539a

5879

2939

6451

2150

7013

3506

4817

4816

5399

2699

5881

2940

6469

924

7019

7018

4831

805

5407

1802

5897

5896

6473

6472

7027

1171

4861

972

5413

2706

5903

5902

6481

270

7039

391

4871

2435

5417

5416

5923

2961

6491

1298

7043

503

4877

1219

5419

5418

5927

5926

6521

8i5

7057

7056

4889

2444

5431

2715

5939

5938

6529

1088

7069

7068

4903

1634

5437

1359

5953

1984

6547

1091

7079

3539

4909

1636

5441

2720

5981

5980

6551

3275

7103

7102

4919

2459

5443

907

5987

2993

6553

6552

7109

7108

4931

4930

5449

2724

6007

858

6563

3281

7121

3560

4933

2466

5471

547

6011

6010

6569

1642

7127

1018

4937

4936

5477

1369

■6029

6028

6571

,6570

7129

594

4943

4942

5479

2739

6037

3018

6577

2192

7151

275

4951

-475

5483

2741

6043

3021

6581

1316

7159

3579

4957

413

5501

5500

6047

6046

6599

3299

7177

7176

4967

4966

5503

5502

6053

3026

6607

2202

7187

3593

4969

828

SS^l

2753

6067

3033

6619

66j8

7193

7192

4973

226

5519

2759

6073

6072

6637

474

7207

7206

4987

2493

5521

345

6079

1013

6653

3326

7211

1030

4993

1664

5527

5526

6089

761

6659

6658

7213

1803

4999

357

5531

5530

6091

2030

6661

6660

7219

7218

5003

2501

SSS7

926

6101

1220

6673

6672

7229

7228

5009

626

5563

2781

6113

6112

6679

3339

7237

402

5011

1670

5569

1392

6121

3060

6689

1672

7243

3621

5021

5020

5573

2786

6131

6130

6691

6690

7247

7246

5023

1674

5581

5580

6133

1533

6701

6700

7253

74

5039

2519

5591

2795

6143

6142

6703

6702

7283

3641

5051

50

5623

5622

6151

1025

6709

6708

7297

2432

5059

5058

5639

2819

6163

79

6719

3359

7307

3653

5077

2538

5641

470

6173

3086

6733

3366

7309

7308

5081

1270

5647

1882

6197

3098

6737

6736

7321

3660

5087

5086

5651

5650

9199

3099

6761

1690

7331

1466

5099

5098

5653

2826

6203

443

6763

161

7333

611

5101

1700

5657

5656

6211

6210

6779

6778

7349

7348

5107

2553

5659

5658

6217

6216

6781

J356

7351

1225

5113

1704

5669

5668

6221

6220

6791

679

7369

1842

5119

853

5683

2841

6229

2076

6793

6792

7393

7392

206

Mr. W. Shanks on the Period of the
Tabie III. (continued).

[Feb. 19,

741 1

7410

7937

7936

8543

8542

9103

9102

9649

603

7417

2472

7949

7948

8563

4281

9109

9108

9661

1380

7433

7432

7951

3975

8573

4286

9127

3042

9677

2419

7451

7450

7963

3981

8581

2860

9133

1522

9679

1613

7457

7456

7993

2664

8597

2149

9137

9136

9689

346

7459

7458

8009

2002

8599

1433

9151

1525

9697

9696

IMl

3738

8011

2670

8609

1076

9157

4578

9719

4859

7481

748

8017

8016

8623

8622

9161

229

9721

4860

7487

7486

8039

4019

8627

4313

9173

4586

9733

2433

7489

1872

8053

4026

8629

2876

9181

3060

9739

9738

7499

7498

8059

8058

8641

4320

9187

4593

9743

9742

7507

3753

8069

8068

8647

8646

9199

4599

9749

9748

IS^l

3758

8081

2020

8663

8662

9203

4601

9767

9766

7523

3761

8087

8086

8669

8668

9209

2302

9769

4884

7529

1882

8089

1348

8677

723

9221

9220

9781

9780

7537

2512

8093

4046

8681

868

9227

4613

9787

4893

7541

7540

8101

1620

8689

2172

9239

4619

9791

4895

7547

3773

8111

811

8693

4346

9241

4620

9803

4901

7549

2516

8117

2029

8699

8698

9257

9256

9811

q8io

7559

3779

8123

4061

8707

4353

9277

4638

9817

9816

7561

1890

8147

4073

8713

8712

9281

. 928

9829

9828

7573

631

8161

1020

8719

4359

9283

1547

9833

9832

ISll

7576

8167

2722

8731

8730

9293

2323

9839

4919

7583

7582

8171

8170

8737

2912

9311

4655

9851

9850

7589

1084

8179

8178

8741

8740

9319

4659

9857

9856

7591

3795

8191

1365

8747

4373

9323

4661

9859

3286

7603

1267

8209

4104

8753

8752

9337

3112

9871

4935

7607

7606

8219

8218

876-i

876

9341

9340

9883

4941

7621

508

8221

2740

8779

22

9343

9342

9887

9886

7639

3819

8231

4115

8783

8782

9349

3116

9901

12

7643

3821

8233

8232

8803

1467

9371

9370

9907

4953

7649

1912

8237

41T8

8807

8806

9377

9376

9923

4961

7669

284

8243

4121

8819

8818

9391

4695

9929

1241

7673

7672

8263

8262

8821

8820

9397

81

9931

9930

7681

1920

8269

8268

8831

4415

9403

1567

9941

1988

7687

7686

8273

8272

8837

4418

9413

4706

9949

9948

7691

7690

8287

8286

8839

4419

9419

554

9967

9966

7699

7698

8291

8290

8849

553

9421

9420

9973

554

7703

7702

8293

2073

8861

8860

9431

4715

10007

iooc6

7717

1929

8297

8296

8863

8862

9433

1048

10009

5004

7723

1287

8311

4155

8867

4433

9437

4718

10037

386

7727

7726

8317

462

8887

8886

9439

1573

10039

5019

7741

860

8329

1041

8893

2223

9461

9460

10061

10060

7753

7752

8353

8352

8923

1487

9463

3154

10067

5033

7757

1939

8363

4181

8929

144

9467

4733

10069

10068

7759

3879

8369

4184

8933

2233

9473

9472

10079

5039

7789

2596

8377

8376

8941

2980

9479

4739

10091

10090

7793

7792

8387

599

8951

4475

9491

9490

10093

2523

7817

7816

8389

8388

8963

4481

9497

9496

10099

3366

7823

7822

8419

2806

8969

4484

9511

1585

10103

10102

7829

7828

8423

8422

8971

8970

9521

595

lOIII

5055

7841

56

8429

8428

8999

4499

9533

2383

10133

2533

7853

3926

8431

4215

9001

1125

9539

9538

10139

10138

7867

3933

8443

4221

9007

3002

9547

4773

10141

10140

7873

7872

8447

8446

9011

9010

9551

955

10151

5075

7877

3938

8461

2820

9013

2253

9587

4793

10159

5079

7879

3939

8467

4233

9029

9028

9601

4800

10163

5081

7883

3941

8501

8500

9041

1130

9613

267

10169

5084

7901

7900

8513

8512

9043

4521

9619

3206

10177

10176

7907

3953

8521

710

9049

4524

9623

9622

10181

10180

7919

3959

8527

2842

9059

9058

9629

9628

10193

10J92

7927

7926

8537

8536

9067

4533

9631

4815

10211

10210

7933

3966

8539

2846

9091

10

9643

4821

I0223

10222

1874.] Reciprocal of every Prime Number belovj 30^000; 207

Table III. {conthmecT).

10243

569

10847

10846

11447

11446

10247

10246

10853

5426

11467

5733

10253

2563

10859

10858

11471

5735

10259

10258

10861

10860

11483

5741

10267

5133

10867

1811

11489

2872

10271

79

10883

5441

11491

766

10273

10272

10889

2722

11497

11496

10289

5144

10891

1210

1 1503

11502

I030I

10300

10903

10902

11519

5759

10303

3434

10909

1212

11527

3842

I03I3

10312

10937

10936

11549

11548

I032I

2580

10939

10938

11551

1925

I033I

2066

10949

10948 1

1 1579

11578

10333

5166

10957

2739

11587

1931

10337

10336

10973

2743

11593

11592

10343

10342

10979

10978

11597

5798

10357

5178

10987

5493

1 1617

11616

10369

2592

10993

10992

11621

11620

I039I

5195

11003

5501

11633

11632

10399

1733

11027

5513

11657

11656

10427

5213

1 1 047

11046

11677

5838

10429

948

11057

11056

11681

5840

10433

10432

0059

11058

11689

487

10453

5226

11069

11068

11699

11698

10457

10456

11071

615

11701

11700

10459

10458

11083

1847

11717

2929

10463

10462

11087

482

11719

5859

, 10477

1746

1 1093

2773

11731

11730

10487

10486

11113

3704

11743

11742

10499

1 049 8

11117

2779

11777

11776

I050I

3500

11119

5559

11779

3926

I05I3

10512

11131

11130

11783

11782

10529

5264

11149

11148

11789

11788

I053I

10530

11159

5579

11801

2950

10559

5279

11161

310

11807

11806

10567

10566

11171

III 70

11813

5906

10589

10588

11173

5586

11821

1 1 820

10597

5298

11177

11176

11827

5913

I060I

1060

11197

2799

11831

169

10607

10606

11213

2803

11833

11832

I06I3

758

11239

5619

11839

5919

10627

5313

1 1243

5621

11863

11862

I063I

5315

11251

2250

11867

5933

10639

5319

11257

11256

11887

11886

I065I

10650

11261

2252

11897

11896

10657

10656

11273

11272

11903

11902

10663

10662

11279

5639

11909

11908

10667

5333

11287

11286

11923

5961

10687

10686

11299

11298

11927

11926

I069I

^0690.

11311

377

11933

5966

10709

10708

11317

943

11939

11938

I07II

595

11321

1132

J1941

11940

10723

5361

11329

1888

11953

11952

10729

596

1135^

5675

1 1959

5979

10733

2683

11353

11352

11969

352

10739

10738

11369

812

11971

11970

10753

3584

11383

11382

11981

11980

I077I

2154

11393

11392

11987

5993

I078I

10780

11399

5699

12007

4002

10789

10788

11411

2282

12011

12010

10799

5399

11423

11422

12037

3009

I083I

5415

1 1437

2859

1 204 1

6020

I0837

63

11443

1907

12043

2007

12049
2071

2073

2097

2IOI
2107
2109
2113
2119
2143
2149

2157
2161

2163

2197

2203

2211

2227

2239

2241

2251

2253
2263

2269

2277

2281

2289

2301

2323
2329

2343

2347

2373

2377

2379
2391

2401

2409

2413

2421

2433

2437

2451

2457

2473
2479

2487

2491

2497
2503
251I

2517
2527

2539

2541

2547

2553
2569

2577

2583
2589

2601

261I

6024

355

12072

4032

12100

6053

4036

121 12

6059

12142

12148

2026

6080

6081

3049
6101

4070

6113
6119
6120

12250

3063
12262

12268

3069
6140

384
2460

6161

3082
4114
6173
6186
12376
12378

6195

1240

6204

6206
1 2420

4144

6218
12450
12456
1 2472

6239
12486
1 2490
1 249 6
12502

2085

149

I2526

12538

4180

6273
12552

6284
12576
12582
12588

6300
12610

12613

12619

12637

12641

12647

12653

12659

12671

12689

12697

12703

12713

12721

12739

12743

12757
12763

12781
12791
12799
12809
12821
12823
12829
12841
12853
12889
12893
12899
12907
12911
12917
12919
12923
12941
12953

12959
12967

12973

12979

12983

13001
13003
13007
13009

13033

13037

13043
13049

13063

13093
13099

13103

13109

13121

13127

13147
13151

13159
13163

13171

13177

J3123

6306
4206

3159
3160

12646

6326

12658

181

793
12696

4234
12712

2120

4246
1274.2

2126

709

12780

6395

2133
6404

12820

12822

4276

6420

459
3222

3223

12898

6453

6455
6458

2153
6461

12940

12952

6479

4322

2162
12978
12982

1625

6501
13006

2168

4344
6518

6521

3262
13062

2182
13098
13102
13108

6560
13126

939

1315
2193

6581

4390
1 3176

4394

208

Mr. W. Shanks on the Period of the [Feb. 19^

Table III, (continued).

I3I87

6593

13807

13806

14461

4820

15061

5020

15629

15628

I32I7

13216

13829

13828

14479

7239

15073

15072

15641

391

I32I9

4406 j

13831

2305

14489

7244

15077

7538

15643

7821

13229

13228

13841

6920

14503

14502

15083

7541

15647

15646

I324I

1655

13859

13858

14519

7259

15091

3018

15649

489

13249

288,

13873

13872

14533

519

15101

604

15661

5220

13259

1894

13877

3469

14537

14536

15107

7553

15667

7833

13267

737

13879

6939

14543

14542

15121

7560

15671

1567

I329I

13290

13883

6941

14549

14548

15131

3026

15679

2613

13297

13296

13901

13900

14551

485

11137

15136

15683

7841

13309

4436

13903

13902

14557

2426

15139

15138

15727

15726

I33I3

13312

13907

409

14561

7280

15149

15148

15731

15730

13327

4442

13913

13912

14563

809

15161

3790

15733

2622

I333I

13330

13921

696

14591

7295

15173

7586

15737

15736

13337

13336

13931

13930

14593

14592

15187

7593

15739

15738

13339

13338

13933

6966

I462I

14620

15193

15192

15749

15748

13367

13366

13963

6981

14627

7313

15199

7599

15761

394

133^1

13380

13967

13966

14629

14628

15217

15216

15767

15766

13397

6698

13997

6998

14633

14632

15227

7613

15773

7886

13399

957

13999

2333

14639

7319

15233

15232

15787

7893

I34II

13410

14009

3502

14653

3663

15241

3810

15791

7895

I34I7

4472

14011

140 10

14657

14656

15259

15258

15797

3949

I342I

13420

14029

14028

14669

14668

15263

15262

15803

7901

I344I

6720

14033

14032

14683

2447

15269

1388

15809

3952

I345I

13450

14051

2810

14699

14698

15271

7635

15817

5272

13457

13456

14057

14056

14713

14712

15277

3819

15823

15822

13463

13462

14071

7035

14717

7358

15287

15286

15859

15858

13469

13468

14081

1760

14723

7361

15289

1274

15877

567

13477

6738

14083

7041

14731

14730

15299

15298

15881

7940

13487

13486

14087

14086

14737

14736

15307

7653

15887

15886

^3499

13498

14107

2351

14741

14740

15313

5104

15889

3972

13513

4504

14143

14142

14747

7373

15319

7659

15901

15900

13523

6761

14149

524

14753

14752

1532-9

7664

15907

7953

13537

13536

14153

14152

14759

7379

15331

51J0

15913

J5912

13553

1936

14159

7079

14767

14766

15349

15348

15919

7959

13567

4522

14173

3543

14771

2110

15359

7679

15923

•7961

13577

13576

14177

14176

14779

14778

15361

256

15937

5312

13591

1359

14197

91

14783

14782

15373

1281

15959

7979

13597

3399

14207

14206

14797

3699

15377

15376

15971

15970

13613

6806

14221

2844

14813

7406

15383

15382

15973

121

13619

1238

14243

7121

I482I

4940

15391

7695

15991

7995

13627

6813

14249

7124

14827

2471

15401

275

16001

2000

13633

4544

14251

14250

14831

1483

15413

7706

16007

16006

13649

853

14281

1190

14843

7421

15427

77^3

16033

16032

13669

13668

14293

1191

14851

990

15439

7719

16057

1784

13679

6839

14303

14302

14867

7433

15443

7721

16061

3212

13681

3420

14321

3580

14869

4956

15451

5150

16063

5354

13687

4562

14323

1023

14879

7439

15461

15460

16067

8033

13691

13690

14327

14326

14887

14886

15467

7733

16069

5356

13693

326

14341

14340

14891

14890

15473

15472

16073

16072

13697

13696

14347

7173

14897

14896

15493

7746

16087

5362

13709

13708

14369

449

14923

7461

15497

1192

16091

16090

13711

6855

14387

7193

14929

1866

15511

7755

16097

16096

13721

6860

14389

4796

14939

14938

15527

15526

16103

16102

13723

6861

14401

3600

14947

7473

15541

740

16111

1611

13729

343^

14407

686

14951

1495

15551

7755

16127

16126

13751

6875

1441 1

2882

14957

7478

11559

2593

16139

16138

13757

362

14419

4806

14969

7484

15569

7784

16141

3228

13759

6879

14423

14422

14983

4994

15581

15580

16183

16182

13763

6881

14431

sss

15013

3753

15583

-^ss^^

16187

8093

13781

13780

14437

218

15017

15016

15601

390

16189

852

13789

13788

14447

14446

15031

7515

15607

15606

16193

16192

13799

6899

14449

3612

15053

3763

15619

15618

16217

16216

1874,] Reciprocal of every Prime Number beloio 20^000. 209

Table III. {continued).

16223
16229
16231
16249
16253
16267
16273
16301
16319

16333

J6339
16349

16361

16363

16369

16381

16411

1 641 7

16421

16427

16433

16447

16451

16453
16477

1 648 1

16487

16493

16519

16529

16547

16553
16561

16567

16573
16603

16607

16619

16631

16633

16649

16651

16657

16661

16673

16691

16693

16699

16703

16729

16741

16747

16759

16763

16787

16811

16823

16829

16831

16843

1 687 1

16879

16883

16222

16228

8115

1354
8126

8133

16272

3260

8159

4083

16338

16348

1636

8181

264

5460

16410

16416

16420

8213

16432

16446

16450

2742

4119

2060

16486

4123

2753
8264

8273
16552

8280
16566

8286

8301
16606
16618

8315

5544
8324

3330
16656
16660
16672
16690

2782
16698
16702

697
16740

8373

931
29

8393
3362

16822

16828

8415

401

1205

2813

8441

6889
6901
6903
6921
6927
6931
6937

6943
6963

6979
6981

6987

6993
701 1

7021

7027

7029

7033
7041

7047

7053
7077

7093
7099

7107

7117

7123

7137

7159
7167

7183

7189
7191

7203

7207

7209

7231

7239

7257
7291

7293
7299

7317

7321

7327

7333

7341

7351

7359

7377

7383

7387

7389

7393
7401

7417

7419

7431

7443

7449
7467

7471
7477

8444
3380

5634
423

5642
16930
16936
16942

8481
16978
16980

8493
16992

17010

3404
8513

17028

17032

2840

5682

1421

2846

4273
17098

8553
4279

8561

5712

8579
17166

17182

17188

8595
2867

17206

4302

8615

8619

17256

17290

1441

1922

3443
8660

17326

8666

5780

8675

2893

17376

17382

8693
17388

17392

4350
17416

17418

8715
8721
4362

8733

8735
8738

7483
7489

7491
7497
7509
7519
7539
755*
7569
7573

7579
7581

7597

7599
7609

7623

7627

7657

7659
7669

7681

7683

7707

7713
7729

7737

7747

7749
7761

7783

7789
7791

7807

7827

7837

7839
7851

7863
7881

7891

7903
7909

7911

7921

7923
7929

7939

7957

7959

7971

7977
7981

7987
7989
8013
8041
8043
8047
8049
8059
8061
8077
8089

8741

2186

17490

17496

5836

8759
5846

2925

2928

4393

17578
17580

4399

2933
8804

1958

8813

17656
5886

17668
1105
8841

8853

17712

iio8

17736

8873
17748

8880
17782
17788

8895
17806

8913

91
8919

714

17862

2235

17890

17902

17908

2985

8960

8961

8964

17938
8978

8979
17970

856

17980

8993

5996

9006

220

291

2578

1128

18058

18060

4519
9044

18097

6032

18II9

9059

I8I2I

2265

I8I27

18126

18131

490

18133

9066

I8I43

I8I42

18149

1396

18169

9084

I8I81

6060

18191

9095

18199

9099

18211

18210

18217

18216

18223

6074

18229

18228

18233

424

18251

18250

18253

9126

18257

2608

18269

18268

18287

18286

18289

9144

18301

18300

18307

3051

18311

9155

I83I3

18312

18329

4582

18341

18340

18353

18352

18367

6l22

18371

3674

18379

18378

18397

9198

18401

4600

^

I84I3

4603

18427

3071

18433

6144

18439

9219

18443

9221

18451

6150

18457

18456

18461

18460

18481

1320

18493

4623

18503

18502

18517

3086

18521

9260

18523

9261

12539

18538

18541

3708

18553

6184

18583

18582

18587

9293

18593

18592

18617

1432

18637

9318

18661

18660

18671

1867

18679

3113

I869I

18690

I870I

3740

I87I3

I8712

8719

8731
8743

8749

8757

8773

8787

8793
8797

8803

8839
8859

8869
8899
8911

8913
8917

8919
8947

8959
8973
8979

9001
9009

9013
9031

9037
905^

9069

9073
9079

9081

9087

9121

9139
9141

9157
9163

9181

9183

9207
9211

9213
9219
9231

9237
9249
9259

9267

9273

9289

9301

9309
9319

9333

9373
9379
9381

9387

9391

9403

9417
9421

9359
3746
18742
18748
3126
9386

9393
18792

127

9401

9419

18858

18868

18898

9455
18912

4729

9459

9473

9479

153
18978

2375
4752

4753

9515
9518

6350
6356

19072

9539

795
6362

9560

19138

19140

4789

9581

19180

6394
19206

3842

9606

19218

9615

4809

3208

19258

9633
19272

4822

3860

19308

9659

4833

4843
19378

1292

3231

1385
9701

19416

1 9420

210

Mr. II. P. Blauford on the

[Feb. 26,

Table III.

(contin

ued) .

19423

3237

19501

780

19681

9840 19777

6952

19913

19912

19427

9713

19507

9753

19687

19686 19793

19792

19919

9959

19429

19428

19531

6510

19697

19696 19801

9900

19927

19926

19433

19432

19541

19540

19699

19698

19813

4953

19937

2848

1 9441

1620

19543

19542

19709

19708

19819

19818

19949

19948

19447

19446

19559

9779

19717

9858

19841

1984

19961

9980

19457

19456

19571

19570

19727

19726

19843

3307

19963

3327

19463

19462

19577

19576

19739

19738

19853

9926

19973

9986

19469

19468

19583

19582

19751

9875

19861

6620

19979

19978

1 947 1

9735

19597

4899

19753

19752

19867

6622

19991

9995

19477

3246

19603

3267

19759

9879

19889

9944

19993

19992

19483

9741

19609

9804

19763

9881 19891

6630

19997

9998

19489

406

19661

19660

February 26, 1874.
JOSEPH DALTON HOOKER, C.B., President, in the Chair
The followiiig Papers were read :—

I. ^^ The Winds of N'orthern India, in relation to the Temperatiir
and Yapour- constituent of the Atmosphere/'' By Heniiy E.
BlajIford, E.G.S., Meteorological Reporter to the Oovernment
of Bengal. Commnnicated by Major-General SxrwiciiEY, 11. E.
Received May 25, 1873.

(Abstract.)

The object of this paper is to describe the normal vvind-currents of
JNorthern India, and their annual variation, and to trace out their origin
and causes, so far as these can be discovered in the local physical
changes of the atmosphere. After referring to the data on which his
conclusions are based, the author goes on to describe the winds of the
principal geographical regions of North India in detail.

Part I. Description of Winds,

1. ThePunjah. — ^As a rule currents from the Vvestward predominate on
an average throughout the year ; and this is also found to be the case in
other parts of North India. In the most northern part of the Punjab,
v/esterly winds prevail in the cold and hot dry months, easterly in the
rainy months. In the central districts northerly winds preponderate
over southerly, having in the cold months a westerly tendency, but
drawing round to the north-east as the hot weather comes on, while as
the rainy season sets in the winds tend to east and south-east, returning
to west after the rain ceases in September. In the southern part of the
Punjab (and this is also the case in Sindh) easterly winds never prevail,